If HOME is removed, then e.g. vim, bash, etc., will use /root/.vimrc, /root/.bashrc, etc rather than the user's ~/.vimrc, ~/.bashrc, etc. While it's a bad idea to run X clients via sudo, they too would likely look in the wrong locations for configuration files, and there's a chance that X11 clients may not even be able to connect to the X11 server if they are aimed at the wrong .Xauthority file.
If HOME is removed, then e.g. vim, bash, etc., will use /root/.vimrc, /root/.bashrc, etc rather than the user's ~/.vimrc, ~/.bashrc, etc. While it's a bad idea to run X clients via sudo, they too would likely look in the wrong locations for configuration files, and there's a chance that X11 clients may not even be able to connect to the X11 server if they are aimed at the wrong .Xauthority file.
Thanks